Topology optimization of beams’ cross-sectional properties considering torsional and warping behavior

This paper introduces a novel efficient topology optimization methodology for beams’ torsion using the warping function formulation. The finite element method is used to discretize the cross-section and an efficient gradient-based optimization problem is formulated to optimize the relevant parameters corresponding to the torsion and warping constants of the beam. As a result, for the first time, one can optimize a beam for problems where the warping behavior is dominant. Density-based optimization is defined where the SIMP approach is utilized to penalize intermediate element densities. A key challenge of the optimization that arises in the warping function framework is the design-dependent nature of the problem. That is, the forcing vector varies during the optimization as it depends on the cross-section boundaries, which are functions of the updating topology. To this end, a differentiable boundary recognition algorithm is proposed. The methodology is applied to design beam cross-sections in which both torsion and warping constants are of interest. While intuitive topologies are obtained in the case of optimized torsion constant, this is not the case for the warping constant. The latter shows unique material distributions and a special dependence on the allowable material density.